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A115584
Number of partitions of n in which each part k occurs more than k times.
12
1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 4, 3, 6, 4, 7, 7, 8, 8, 12, 9, 15, 14, 17, 18, 24, 21, 29, 29, 35, 35, 46, 42, 56, 54, 65, 67, 81, 77, 98, 95, 115, 114, 139, 135, 164, 165, 190, 195, 230, 225, 272, 271, 313, 321, 370, 374, 433, 441, 501, 514, 589, 592, 681, 698, 778, 809, 907
OFFSET
0,7
COMMENTS
The Heinz numbers of these partitions are given by A325127. - Gus Wiseman, Apr 02 2019
LINKS
FORMULA
G.f.: Product_{k>=1} (1-x^k+x^(k*(k+1)))/(1-x^k).
EXAMPLE
a(2) = 1 because we have [1,1]; a(10) = 4 because we have [2,2,2,2,2], [2,2,2,2,1,1], [2,2,2,1,1,1,1] and [1^10].
From Gus Wiseman, Apr 02 2019: (Start)
The initial terms count the following integer partitions:
0: ()
2: (11)
3: (111)
4: (1111)
5: (11111)
6: (222)
6: (111111)
7: (1111111)
8: (2222)
8: (22211)
8: (11111111)
9: (222111)
9: (111111111)
10: (22222)
10: (222211)
10: (2221111)
10: (1111111111)
11: (2222111)
11: (22211111)
11: (11111111111)
12: (3333)
12: (222222)
12: (2222211)
12: (22221111)
12: (222111111)
12: (111111111111)
(End)
MAPLE
g:=product((1-x^k+x^(k*(k+1)))/(1-x^k), k=1..30): gser:=series(g, x=0, 75): seq(coeff(gser, x, n), n=0..70); # Emeric Deutsch, Mar 12 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +add(b(n-i*j, i-1), j=i+1..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..80); # Alois P. Heinz, Feb 09 2017
MATHEMATICA
CoefficientList[ Series[ Product[(1 - x^k + x^(k(k + 1)))/(1 - x^k), {k, 14}], {x, 0, 66}], x] (* Robert G. Wilson v, Mar 12 2006 *)
Table[Length[Select[IntegerPartitions[n], And@@Table[Count[#, i]>i, {i, Union[#]}]&]], {n, 0, 30}] (* Gus Wiseman, Apr 02 2019 *)
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Mar 09 2006
EXTENSIONS
More terms from Robert G. Wilson v and Emeric Deutsch, Mar 12 2006
STATUS
approved